Download Transport Phenomena

Transport Phenomena
Interphase Transport in Isothermal Systems
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Interphase Transport in Isothermal Systems
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Definition of friction factors
Friction factors for flow in tubes
Friction factors for flow around spheres
Friction factors for packed columns
1. Definition of Friction Factors
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We consider the steadily driven flow of a fluid of constant density in one of two systems: (a) the
fluid flows in a straight conduit of uniform cross section; (b) the fluid flows around a submerged
object that has an axis of symmetry parallel to the direction of the approaching fluid. There will
be a force Ff-s exerted by the fluid on the solid surfaces. It is conveninent to split this force nto two
parts:
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Fs - force that would be extered by the fluid event if it were stationary
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Fk - the additional force associated with the motion of the fluid
In systems of type (a),Fk points in the same direction as the average velocity <v> in the conduit, and in
systems of type (b),Fk points in the same direction as the approach velocity v∞. For both types of
systems we state that the magnitude of the force FA.is proportional to a characteristic area A and a
characteristic kinetic energy К per unit volume:
where f is called the friction factor.
This equation is only defintion of f. This useful because the dimensionless quantity f can be given as a
relatively simple function of the Reynolds number and the system shape.
1. Definition of Friction Factors
Clearly, for any given flow system,/is not defined until A and К are specified.
(a) Fоr flow in conduits, A is usually taken to be the wetted surface, and К is taken to
Specifically, for circular tubes of radius R and length L :
Generally, the quantity measured is not Fk, but rather the pressure difference p0 — pL and the elevation
difference h0 —hL. A force balance on the fluid between 0 and L in the direc- tion of flow gives for
fully developed flow
Elimination of Fk between the last two equations then gives
in which D = 2R is the tube diameter. The quantity f is sometimes called the Fanning friction facto
1. Definition of Friction Factors
(b) For flow around submerged objects, the characteristic area A is usually taken to be the area
obtained by projecting the solid onto a plane perpendicular to the velocity of the approaching fluid; the
quantity К is taken to be , where v∞ is the approach velocity of the fluid at a large distance from
the object. For example, for flow around a sphere of radius R, we define f by the equation
If it is not possible to measure Fk, then we can measure the terminal velocity of the sphere when it falls
through the fluid (in that case, v∞ has to be interpreted as the terminal velocity of the sphere). For the
steady-state fall of a sphere in a fluid, the force Fk is just counterbalanced by the gravitational force on
the sphere less the buoyant force:
Elimination of Fk between the last two equations then gives
This expression can be used to obtain / from terminal velocity data
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2. Friction Factors For Flow In Tubes
„Test section" of inner radius R and length L, shown in picture,
carrying a fluid of constant density and viscosity at a steady
mass flow rate.The pressures P0 and PL the ends of the test
section are known. The system is either in steady laminar flow
or steadily driven turbulent flow (i.e., turbulent flow with a
steady total throughput). In either case the force in the z
direction of the fluid on the inner wall of the test section is
Section of a cirular pipe from z = 0 to z = L for the discussion of dimensional
analysis
In turbulent flow the force may be a function of time, not only because of the turbulent fluctuations, but also because of
occasional ripping off of the boundary layer from the wall, which results in some distances with long time scales. In laminar
flow it is understood that the force will be independent of time.
we can introduce the dimensionless quantities from chapter 3.7:
so we have:
2. Friction Factors For Flow In Tubes
This relation is valid for laminar or turbulent flow in circular tubes. We see that for flow systems in which the drag depends on viscous forces
alone („form drag") the product of fRe is essentially a dimensionless velocity gradient averaged over the surface.
Boundary conditions:
That is, the functional dependence of v and P must, in general, include all the dimensionless variables and the one dimensionless group
appearing in the differential equations. No additional dimensionless groups enter via the preceding boundary conditions. As a consequence,
∂vz/∂r must likewise depend on r,0,z, t, and Re. When ∂vz/∂r dr is evaluated at r = 1/2 and then integrated over z and 0 in equation of the top, the
result depends only on t, Re, and L/D (the latter appearing in the upper limit in the integration over z). Therefore we are led to the conclusion that
f(t) = /(Re, L/D, t), which, when time averaged, becomes
when the time average is performed over an interval long enough to include any long- time turbulent disturbances. The measured friction factor
then depends only on the Reynolds number and the length-to-diameter ratio.
2. Friction Factors For Flow In Tubes
The laminar curve on the friction factor chart is merely a plot of the Hagen-Poiseuille equation.
2. Friction Factors For Flow In Tubes
Turbulent curves have been constructed by using experimental data. Some analytical curve-fit expressions are also available. For turbulent flow in
noncirculartubes it is common to u se the following empiricism: First we define a "mean hydraulic radius" Rh as follows:
In which S is the cross section of the conduit and Z is the wetted perimeter.
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we get:
This estimation method is not for laminar flow!!!
2. Friction Factors For Flow In Tubes
Example
What pressure gradient is required to cause diethylaniline, C6H5N(C2H5)2, to flow in a horizontal, smooth, circular tube of
inside diameter D = 3 cm at a mass rate of 1028 g/s at 20℃? At this temperature the density of diethylaniline is 0.935 g/cm3
and its viscosity is µ = 1.95
Solution
The Reynolds number for the flow is
We find from the table that for this Reynolds number the friction factor f has a value of 0.0063 for smooth tubes. Hence the
pressure gradient required to maintain the flow is
3. Friction Factors For Flow Around Spheres
Recall from chapter 2.6 that the total force acting in the z direction on thesphere can be written as the sum of a contribution from the normal
stresses (Fn) and one from the tangential stresses (Ft). One part of the normal-stress contribution is the force that would be present even if the
fluid were stationary, Fs.Thus the "kinetic force," associated withthe fluid motion,is
The forces associated with the form drag and the friction drag are then obtained from
Since vr is zero everywhere on the sphere surface, the term containing dvr/d0 is zero. If now we split f into two parts as follows
then, from the definition we get:
3. Friction Factors For Flow Around Spheres
The friction factor and Reynolds number is expressed here in terms of dimensionless variables
We are defining the boundary conditions:
Because no additional dimensionless groups enter via the boundary and initial conditions, we know that the dimensionless pressure and velocity
profiles will have the following form:
Tube flow and flow arround a sphere behave quite fifferently. Several points of difference between the two systems are:
Flow in Tubes
Flow Around Spheres
- Rather well defined laminar-turbulent transition at about Re = 2100 - No well defined laminar-turbulent transition
- The only contribution to f is the friction drag
- Contributions to f from both friction and form drag
- No boundary layer separation
- There is a kink in the f vs. Re curve associated with a shift in the
separation zone
3. Friction Factors For Flow Around Spheres
For the creeping flow region, the drag force is given by Stokes' law, which is a consequence of solving the continuity equation and the NavierStokes equation of motion without the ϱDv/Dt term. Stokes' law can be rearranged:
Hence for creeping flow around a spher
and this is the straight-line asymptote as Re —> 0 of the friction factor curve in picture.
Friction factor (or drag coefficient) for spheres moving relative
to a fluid with a velocity v∞.
3. Friction Factors For Flow Around Spheres
Example
Glass spheres of density ρsph = 2.62 g/cm3 are to be allowed to fall through liquid CC14 at 20°C in an experiment for studying human reaction
times in making time observations with stopwatches and more elaborate devices. At this temperature the relevant properties of CC14 are ρ = 1.59
g/cm3 and µ = 9.58 millipoises. What diameter should the spheres be to have a terminal velocity of about 65 cm/s?
Solution
To find the sphere diameter, we have to solve previous equation for D. However, in this equation one has to know D
in order to get f; and f is given by the solid curve in picture. A trial-and-error procedure can be used, taking f = 0.44
as a first guess.
But anothar way is by solving previous equation for f and then note that f/Re is a quantity independet of D we have:
The quantity on the right side can be calculated with the information above, and we call it С Hence we have two
simultaneous equations to solve:
a
b
First equation is a straight line with slope of unity on the log-log plot of f versus Re. For the problem at hand we have
Hence at Re = 105, according to. Eq. a, f = 1.86. The line of slope 1 passing through f = 1.86 at Re = 105 is shown in Picture. This line
intersects the curve of Eq. b (the curve of picture.) at Re = Dv∞ρ/µ = 2.4 x 104. The sphere diameter is then found to be